Synchronization by Nonlinear Frequency Pulling

Abstract

We analyze a model for the synchronization of nonlinear oscillators due to reactive coupling and nonlinear frequency pulling motivated by the physics of arrays of nanoscale oscillators. We study the model for the mean field case of all-to-all coupling, deriving results for the onset of synchronization as the coupling or nonlinearity increase, and the fully locked state when all the oscillators evolve with the same frequency.PACS numbers: 85.85.+j, 05.45.Xt, 62.25.+g In the last decade we have witnessed exciting technological advances in the fabrication of nanoelectromechanical systems (NEMS). Such systems are being developed for a host of nanotechnological applications, as well as for basic research in the mesoscopic physics of phonons and the general study of the behavior of mechanical degrees of freedom at the interface between the quantum and the classical worlds [1,2]. Among the outstanding features of nanomechanical resonating elements is the fact that at these dimensions their normal frequencies are extremely high-recently exceeding the 1GHz mark [3]-facilitating the design of ultra-fast mechanical devices. Since with diminishing size output signals diminish as well, there is a need to use the coherent response in large arrays of coupled nanomechanical resonators (like the ones that were recently fabricated [4,5]) for signal enhancement and noise reduction. One potential obstacle for achieving such coherent response is the fundamental problem of the irreproducibility of NEMS devices. Clearly, as the size of a resonating beam or cantilever decreases to the point that its width is only that of a few dozen atoms, any misplaced atomic cluster dramatically can change the normal frequency or any other property of the resonator. Thus, it is almost inevitable that an array of nanomechanical resonators will contain a distribution of normal frequencies. Here we propose to overcome this potential difficulty by making use of another typical feature of nanomechanical resonators-their tendency to behave nonlinearly at even modest amplitudes. We shall demonstrate here that systems of coupled nonlinear nanomechanical resonators (like the one we studied recently [6]) can self-synchronize to one common frequency through the dependence of their frequencies on the amplitude of oscillation.The synchronization of systems of coupled oscillators that have a distribution of individual frequencies is important in many disciplines of science [7,8]. The coherent oscillations can be used to enhance the sensitivity of detectors or the power output from sources, as proposed here. Synchronization is also important in biological phenomena, for example the collective behavior in populations of animals, such as the synchronized flashing of fire flies, and the coherent oscillations observed in the brain.Although synchronization is often put forward as an example of the importance of understanding a nonlinear phenomenon, the intuition for the phenomenon, and indeed the subsequent mathematical discussion, can often be developed in terms of ...

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